Ok, so I have researched it a bit futher...
Many SEO tools provide some kind of link research options. In most cases it’s just a list of links, type of link and anchor text. They usually do not allow uploading your own list for analysis.
The 3 biggest ones are:
MOZ Open Site Explorer
https://moz.com/researchtools/ose/
They have their own link discovery bots. No possibility to upload a list.
Data: Title, URL, anchor text, spam score, page authority, domain authority
Ahrefs
https://ahrefs.com/
They have their own link discovery bots. No possibility to upload a list.
Data: they categorize links (follow/nofollow, edu/org, text/image, etc.), provide anchor text and assess the backlink strength.
Majestic
https://majestic.com/
Source of data unknown (probablly have their own bots). It is possible to upload URLs for crawling.
Data: URL, anchor text, trust flow, citation flow, link type, date discovered.
I haven’t found any tool that crawls the source code of the backlinking page beyond the anchor text. So, they do not really provide any context to how your site is actually linked.